Homework #3
Math 6070-1, Spring 2006
1. Consider a probability density kernel K of the form
K(x) =
1 −|x|/τ
e
,
2τ
−∞ < x < ∞.
Here, τ > 0 is fixed. Assuming that f is sufficiently smooth, then derive
the form of the asymptotically optimal bandwidth hn in the same manner
as we did in the lectures for the case τ = 1. The extra parameter τ is
often used to refine kernel-density estimates that are based on the doubleexponential family.
2. Construct continuous probability densities f1 , f2 , . . . and f such that:
R∞
(a) limn→∞ −∞ |fn (x) − f (x)| dx = 0; and
(b) there exist infinitely-many values of x such that fn (x) 6→ f (x) as
n → ∞.
Thus, convergence in L1 is not the same as ordinary (pointwise) convergence.
3. Prove that if f and g are probability densities, then F (f ∗g)(t) = (F f )(t)×
(F g)(t) for all t. Use this to prove that if f is a probability density and
φ is the N (0 , 2 ) density, then f ∗ φ has an integrable Fourier transform.
4. Let X1 , X2 , . . . be an i.i.d. sample from a density function f . We assume
that f is differentiable in an open neighborhood V of a fixed point x, and
B := maxz∈V |f 0 (z)| < ∞.
(a) Prove that for all λ > 0, m ≥ 1, and all x ∈ R,
#m
"
Z x+λ
P min |Xj − x| ≥ λ = 1 −
f (z) dz
.
1≤j≤m
x−λ
(b) Prove that for all > 0 small enough,
f (x) − f (z) ≤ 2B.
max
z∈[x− ,x+]
Use this to estimate |
R x+
x−
f (z) dz − 2f (x)|.
(c) Suppose that as m → ∞, λm → ∞ and λ2m /m → 0. Then, prove
that
λm
−1
ln P min |Xj − x| ≥
= f (x).
lim
m→∞ 2λm
1≤j≤m
m
(d) Devise an estimator of f (x) based on the previous steps.